Generalized Josephson effect with arbitrary periodicity in quantum magnets

Easy-plane quantum magnets are strikingly similar to superconductors, allowing for spin supercurrent and an effective superconducting phase stemming from their <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>U</a:mi><a:mo>(</a:mo><a:mn>1</a:mn><a:mo>)</a:mo></a:mrow></a:math> rotation symmetry around the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>z</b:mi></b:math> axis. We uncover a generalized fractional Josephson effect with a periodicity that increases linearly with system size in one-dimensional spin-<c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mn>1</c:mn><c:mo>/</c:mo><c:mn>2</c:mn></c:mrow></c:math> chains at selected anisotropies and phase-fixing boundary fields. The effect combines arbitrary integer periodicities in a single system, exceeding the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mn>4</d:mn><d:mi>π</d:mi></d:mrow></d:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mn>8</e:mn><e:mi>π</e:mi></e:mrow></e:math> periodicity of superconducting Josephson effects of Majorana zero modes and other exotic quasiparticles. We reveal a universal energy-phase relation and connect the effect to the recently discovered phantom helices.